21 PRODUCT OF MEASURES 237
have
k = p
<; 1 for all / e B, and therefore
j.
Minorize this integral with the help of (b).) (Radernacher-KolmogorofT theorem.)
13. Let X be a locally compact space, JLC a positive measure on X, and (un)n > 0 a sequence of
/x-integrable complex functions such that, as H runs through the set of finite subsets of
N, the set of numbers ]T un(x) dp,(x) is bounded. Show that in these conditions the
J \n eH
oo
series ]T \un(x)\2 is convergent almost everywhere in X. (Observe that, for all
n = 0
t e I = [0, 1 ], the set of numbers ^ uk(x)rk(t) d/x(x) is bounded above by a number
independent of n and t\ then use Problem 10(f) and the Lebesgue-Fubini theorem.)
14. Let X be a compact space, ^ a positive measure on X, and (/„) an orthonormal
sequence in «£?c(X, /*) which is uniformly bounded in X.
(a) Let (bn) be a sequence of real numbers such that ^ b% = +00. Show that it is
nsl
oo
not possible that £ \bn(g |/w)| < + oo for all continuous functions g on X. (Argue by
n= 1
contradiction : the functions un(x) = bnfn(x) would satisfy the conditions of Prob-
lem 13, by virtue of the Banach-Steinhaus theorem applied to the linear forms
0H+2W0I/-) on <f(X). Hence f>fl2|/n(*)l2 < +°° almost everywhere in X.
neH it=l
Using EgorofT's theorem, the fact that the functions /„ are uniformly bounded and
/• 00
l/nW|2 djj,(x) = 1, obtain a contradiction of the hypothesis V 6J = +00.)
J 11=1
(b) Let q be a real number such that 1 < q < 2. Show that it is not possible that
oo
2 \(9 \fn)\9 < -H oo for all continuous functions g on X. (If p = q\(q — 1), show that
if the assertion were false there would exist a sequence (bn) of real numbers > 0
such that 2 W ^ + °°» S W < + °° and S I W# !/«)! < + °° for a11 continuous
functions g.)
(c) Deduce from (b) that there exists a continuous function g on X such that, for all
q satisfying 1 < q < 2, we havej) |(# |/B)|« = +00. (Use the principle of condensation
n
of singularities (Section 12.16, Problem 14).)
15. On the space R", let A be Lebesgue measure and let ||jc|| be a norm for which the unit
ball ||je|| <J 1 has measure 1. Let (#*)*>! be an infinite sequence of distinct points
in an integrable bounded set B such that A(B) = 1. For each integer m> 1, let
dm denote the smallest of the numbers ||#i — #/|| with 1 ^i<j<>m. Show that
lim inf m dm ^ c^ \ whereunless / is of the form k - 2"" with k integral and 0 < k < 2".